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arXiv:2006.14152 · Physical review. B./Physical review. B

Quantum-critical scaling properties of the two-dimensional random-singlet state

Lu Liu, Wenan Guo, Anders W. Sandvik

Published Aug 31, 2020• 18 citations
arXivHTMLPDF

Abstract

Using quantum Monte Carlo simulations, we study effects of disorder on the $S=1/2$ Heisenberg model with exchange constant $J$ on the square lattice supplemented by multispin interactions $Q$. It was found recently [L. Liu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this $J\ensuremath{-}Q$ model with random couplings undergoes a quantum phase transition from the N\'eel antiferromagnetic state into a randomness-induced spin-liquid-like state that is a close analog to the well known random-singlet (RS) state of the Heisenberg chain with random couplings. This 2D RS state arises from a spontaneously symmetry-broken fourfold degenerate columnar valence-bond solid that is broken up by the disorder into finite domains, with spinons localized at topological defects. The interacting spinons form a critical collective many-body state without magnetic long range order but with the mean spin-spin correlations decaying with distance $r$ as ${r}^{\ensuremath{-}2}$, as in the one-dimensional RS state. The dynamic exponent $z\ensuremath{\ge}2$, varying continuously with the model parameters. In this work, we further investigate the properties of the RS state in the $J\ensuremath{-}Q$ model with random $Q$ couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit. We also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with the conventional quantum-critical scaling laws when the condition implied by the ${r}^{\ensuremath{-}2}$ form of the spin correlations is imposed. In particular, all the different quantities can be explained by the same value of the dynamic exponent $z$ at fixed model parameters. We argue that the RS state identified in the $J\ensuremath{-}Q$ model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings and that it has already been observed experimentally.

DOI
10.1103/physrevb.102.054443

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Using quantum Monte Carlo simulations, we study effects of disorder on the S=1/2S=1/2S=1/2 Heisenberg model with exchange constant JJJ on the square lattice supplemented by multispin interactions QQQ. It was found recently [L. Liu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this J\ensuremath−QJ\ensuremath{-}QJ\ensuremath−Q model with random couplings undergoes a quantum phase transition from the N'eel antiferromagnetic state into a randomness-induced spin-liquid-like state that is a close analog to the well known random-singlet (RS) state of the Heisenberg chain with random couplings. This 2D RS state arises from a spontaneously symmetry-broken fourfold degenerate columnar valence-bond solid that is broken up by the disorder into finite domains, with spinons localized at topological defects. The interacting spinons form a critical collective many-body state without magnetic long range order but with the mean spin-spin correlations decaying with distance rrr as r\ensuremath−2{r}^{\ensuremath{-}2}r\ensuremath−2, as in the one-dimensional RS state. The dynamic exponent z\ensuremath≥2z\ensuremath{\ge}2z\ensuremath≥2, varying continuously with the model parameters. In this work, we further investigate the properties of the RS state in the J\ensuremath−QJ\ensuremath{-}QJ\ensuremath−Q model with random QQQ couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit. We also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with the conventional quantum-critical scaling laws when the condition implied by the r\ensuremath−2{r}^{\ensuremath{-}2}r\ensuremath−2 form of the spin correlations is imposed. In particular, all the different quantities can be explained by the same value of the dynamic exponent zzz at fixed model parameters. We argue that the RS state identified in the J\ensuremath−QJ\ensuremath{-}QJ\ensuremath−Q model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings and that it has already been observed experimentally.
Quenched disorders can strongly influence the physical properties of quantum many-body systems. The real-space strong-disorder renormalization group (SDRG) analysis has shown that the spin-1/2 random Heisenberg chain is controlled by the infinite-randomness fixed point (IRFP) and forms a random singlet (RS) ground state. Motivated by recent thermal transport experiments on the quasi-one-dimensional antiferromagnet copper benzoate [B. Y. Pan et al., Phys. Rev. Lett. 129, 167201 (2022)], we adapt the SDRG to study the low-temperature properties of the random Heisenberg chain by assuming that its low-energy excited states are captured by the parent Hamiltonian of the RS ground state as well. We find that while the specific heat coefficient and the uniform magnetic susceptibility scale as $C/T\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{c}}$ and $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{s}}$ with $0<{\ensuremath{\alpha}}_{c,s}<1$, indicating a divergent low-energy density of states, the thermal and the spin conductivities scale as $\ensuremath{\kappa}/T\ensuremath{\sim}T$ and ${\ensuremath{\sigma}}_{s}\ensuremath{\sim}T$, which implies a vanishing density of extended states in the low-energy limit. We believe that such a disparity in the thermodynamic and transport properties is a common feature of random systems controlled by the IRFP.
Quenched randomness strongly affects properties of magnetic materials. Two-dimensional (2D) random singlet (RS) states emerge in the random $J\text{\ensuremath{-}}Q$ model by destroying valence bond solid order with spatial randomness. We examine the 2D RS states in a magnetic field with quantum Monte Carlo simulations. The magnetization and susceptibilities show power-law scaling with magnetic field at low temperature. Moreover, they show one-parameter scaling behavior with B/TB/TB/T, and the scaling functions are remarkably consistent with a phenomenological model of random spin pairs with a singular distribution of interactions. These universal scaling functions can be used to diagnose 2D RS states in experiments.
We resolve the nature of the quantum phase transition between antiferromagnetic and valence-bond solid ground states in square-lattice spin-1/2 magnets - the putative deconfined quantum-critical point. Using quantum Monte Carlo simulations of J-Q models, in which Heisenberg exchange J competes with multi-spin interactions Qn formed by n singlet projectors, we demonstrate first-order discontinuities increasing with n. For n=2 and n=3, the first-order signatures are very weak and we extract universal scaling dimensions of a proximate multicritical point. Their values coincide closely with known bounds for an SO(5) symmetric conformal field theory (CFT) containing a relevant SO(5) singlet operator that destabilizes the generic deconfined critical point. We characterize the emergent SO(5) symmetry by its scaling dimensions of the leading irrelevant perturbations. Using the J-Q2-Q6 model, we tune the transition from weak to stronger first-order and find scaling of the coexisting order parameters with a large exponent beta ~ 0.85, in agreement with the relevant scaling dimensions of the multicritical point. The exponent that had previously been identified with the correlation length does not have a corresponding CFT level, and we explain it as pseudo-critical length scale of bubbles of one phase inside the other that are compressed in size by an SO(5)-violating irrelevant repulsive interaction between the two order parameters. The bubble scale reflects a hierarchy of excitation gaps generated by the SO(5) perturbation in the neighborhood of the first-order transition, akin to a Higgs mechanism. We argue that the multicritical point is the tip of a gapless spin liquid phase in frustrated Heisenberg models into which the J-Q models can be deformed, but the spin liquid, and perhaps the multicritical point itself, exist outside the regime amenable to sign-free simulations.
We use a tensor network renormalization group method to study random S=2S=2S=2 antiferromagnetic Heisenberg chains with alternating bond strength distributions. In the absence of randomness, an imposed dimerization with bond alternation induces two quantum critical points between the S=2S=2S=2 Haldane phase, a partially dimerized phase, and a fully dimerized phase, depending on the strength of dimerization. These three phases, called ($\ensuremath{\sigma},4\ensuremath{-}\ensuremath{\sigma}$) = (2,2), (3,1), and (4,0) phases, are valence-bond solid (VBS) states characterized by $\ensuremath{\sigma}$ valence bonds (effective spin-1/2 singlet pairs) forming across even links and $4\ensuremath{-}\ensuremath{\sigma}$ valence bonds on odd links. Here we study the effects of bond randomness on the ground states of the dimerized spin chain, calculating disorder-averaged twist order parameters and spin correlations. We classify the types of random VBS phases depending on the strength of bond randomness RRR and dimerization DDD using the twist order parameter, which has a negative/positive sign for a VBS phase with odd/even $\ensuremath{\sigma}$. Our results demonstrate the existence of a multicritical point in the intermediate disorder regime with finite dimerization, where (2,2), (3,1), and (4,0) phases meet. This multicritical point is at the junction of three phase boundaries in the $R\text{\ensuremath{-}}D$ plane: the (2,2)-(3,1) and (3,1)-(4,0) boundaries that extend to zero randomness, and the (2,2)-(4,0) phase boundary that connects another multicritical point in the undimerized limit. The undimerized multicritical point separates a gapless Haldane phase and an infinite-randomness critical line with the diverging dynamic critical exponent in the large-RRR limit at D=0D=0D=0. Furthermore, we identify the (3,1)-(4,0) phase boundary as an infinite-randomness critical line even at small RRR and find the signature of infinite randomness at the (2,2)-(3,1) phase boundary only in the vicinity of the multicritical point.
Understanding disorder-induced exotic states in two-dimensional magnets is one of the central topics in the study of frustrated magnetism. Here, the authors focus on the randomness-induced disordered phase in the spin-\textonehalf{} ${J}_{1}\ensuremath{-}{J}_{2}$ square-lattice Heisenberg model with a strong bond randomness, by using large-scale density matrix renormalization group (DMRG) simulations. The results not only reveal the absent spin-glass order, but also identify the power-law decay of the average spin correlation and the exponential decay of the typical spin correlation. These all agree with the corresponding behavior in the one-dimensional random-singlet state and characterize the random-singlet nature of this disordered phase. This shows that DMRG simulations provide opportunities for exploring the disorder effect in two-dimensional magnets.
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